\(\int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx\) [57]

Optimal result

Integrand size = 21, antiderivative size = 91 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^3(c+d x)}{3 a d}-\frac {3 \cos ^5(c+d x)}{5 a d}+\frac {3 \cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 a d}+\frac {\sin ^8(c+d x)}{8 a d} \]

[Out]

1/3*cos(d*x+c)^3/a/d-3/5*cos(d*x+c)^5/a/d+3/7*cos(d*x+c)^7/a/d-1/9*cos(d*x+c)^9/a/d+1/8*sin(d*x+c)^8/a/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2914, 2644, 30, 2645, 276} \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^8(c+d x)}{8 a d}-\frac {\cos ^9(c+d x)}{9 a d}+\frac {3 \cos ^7(c+d x)}{7 a d}-\frac {3 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]

[In]

Int[Sin[c + d*x]^9/(a + a*Sec[c + d*x]),x]

[Out]

Cos[c + d*x]^3/(3*a*d) - (3*Cos[c + d*x]^5)/(5*a*d) + (3*Cos[c + d*x]^7)/(7*a*d) - Cos[c + d*x]^9/(9*a*d) + Si
n[c + d*x]^8/(8*a*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^9(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \sin ^7(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^7(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int x^7 \, dx,x,\sin (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\sin ^8(c+d x)}{8 a d}+\frac {\text {Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos ^3(c+d x)}{3 a d}-\frac {3 \cos ^5(c+d x)}{5 a d}+\frac {3 \cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 a d}+\frac {\sin ^8(c+d x)}{8 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {(4258+6995 \cos (c+d x)+3650 \cos (2 (c+d x))+1085 \cos (3 (c+d x))+140 \cos (4 (c+d x))) \sin ^{10}\left (\frac {1}{2} (c+d x)\right )}{315 a d} \]

[In]

Integrate[Sin[c + d*x]^9/(a + a*Sec[c + d*x]),x]

[Out]

((4258 + 6995*Cos[c + d*x] + 3650*Cos[2*(c + d*x)] + 1085*Cos[3*(c + d*x)] + 140*Cos[4*(c + d*x)])*Sin[(c + d*
x)/2]^10)/(315*a*d)

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {\cos \left (d x +c \right )^{8}}{8}+\frac {3 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}-\frac {3 \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\)	\(89\)
default	\(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {\cos \left (d x +c \right )^{8}}{8}+\frac {3 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}-\frac {3 \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\)	\(89\)
parallelrisch	\(\frac {8820 \cos \left (4 d x +4 c \right )+17640 \cos \left (d x +c \right )+27409+315 \cos \left (8 d x +8 c \right )+900 \cos \left (7 d x +7 c \right )-2520 \cos \left (6 d x +6 c \right )-2016 \cos \left (5 d x +5 c \right )-17640 \cos \left (2 d x +2 c \right )-140 \cos \left (9 d x +9 c \right )}{322560 d a}\)	\(96\)
norman	\(\frac {\frac {32}{315 a d}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d a}+\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d a}+\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d a}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\)	\(121\)
risch	\(\frac {7 \cos \left (d x +c \right )}{128 a d}-\frac {\cos \left (9 d x +9 c \right )}{2304 a d}+\frac {\cos \left (8 d x +8 c \right )}{1024 a d}+\frac {5 \cos \left (7 d x +7 c \right )}{1792 a d}-\frac {\cos \left (6 d x +6 c \right )}{128 a d}-\frac {\cos \left (5 d x +5 c \right )}{160 a d}+\frac {7 \cos \left (4 d x +4 c \right )}{256 a d}-\frac {7 \cos \left (2 d x +2 c \right )}{128 a d}\)	\(135\)

[In]

int(sin(d*x+c)^9/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(-1/9*cos(d*x+c)^9+1/8*cos(d*x+c)^8+3/7*cos(d*x+c)^7-1/2*cos(d*x+c)^6-3/5*cos(d*x+c)^5+3/4*cos(d*x+c)^4+
1/3*cos(d*x+c)^3-1/2*cos(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 1080 \, \cos \left (d x + c\right )^{7} + 1260 \, \cos \left (d x + c\right )^{6} + 1512 \, \cos \left (d x + c\right )^{5} - 1890 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 1260 \, \cos \left (d x + c\right )^{2}}{2520 \, a d} \]

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(280*cos(d*x + c)^9 - 315*cos(d*x + c)^8 - 1080*cos(d*x + c)^7 + 1260*cos(d*x + c)^6 + 1512*cos(d*x +
c)^5 - 1890*cos(d*x + c)^4 - 840*cos(d*x + c)^3 + 1260*cos(d*x + c)^2)/(a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**9/(a+a*sec(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 1080 \, \cos \left (d x + c\right )^{7} + 1260 \, \cos \left (d x + c\right )^{6} + 1512 \, \cos \left (d x + c\right )^{5} - 1890 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 1260 \, \cos \left (d x + c\right )^{2}}{2520 \, a d} \]

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/2520*(280*cos(d*x + c)^9 - 315*cos(d*x + c)^8 - 1080*cos(d*x + c)^7 + 1260*cos(d*x + c)^6 + 1512*cos(d*x +
c)^5 - 1890*cos(d*x + c)^4 - 840*cos(d*x + c)^3 + 1260*cos(d*x + c)^2)/(a*d)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.55 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {32 \, {\left (\frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {126 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {630 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1\right )}}{315 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \]

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

32/315*(9*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 36*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 84*(cos(d*x +
c) - 1)^3/(cos(d*x + c) + 1)^3 - 126*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 630*(cos(d*x + c) - 1)^5/(co
s(d*x + c) + 1)^5 - 1)/(a*d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^2}{2\,a}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}-\frac {3\,{\cos \left (c+d\,x\right )}^4}{4\,a}+\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a}+\frac {{\cos \left (c+d\,x\right )}^6}{2\,a}-\frac {3\,{\cos \left (c+d\,x\right )}^7}{7\,a}-\frac {{\cos \left (c+d\,x\right )}^8}{8\,a}+\frac {{\cos \left (c+d\,x\right )}^9}{9\,a}}{d} \]

[In]

int(sin(c + d*x)^9/(a + a/cos(c + d*x)),x)

[Out]

-(cos(c + d*x)^2/(2*a) - cos(c + d*x)^3/(3*a) - (3*cos(c + d*x)^4)/(4*a) + (3*cos(c + d*x)^5)/(5*a) + cos(c +
d*x)^6/(2*a) - (3*cos(c + d*x)^7)/(7*a) - cos(c + d*x)^8/(8*a) + cos(c + d*x)^9/(9*a))/d